Preliminary Exam - Summer 1979

Problem 1   Prove that the matrix

$$\left( \begin{array}{cccc} 0 & 5 & 1 & 0 \\ 5 & 0 & 5 & 0 \\ 1 & 5 & 0 & 5 \\ 0 & 0 & 5 & 0\end{array} \right)$$

has two positive and two negative eigenvalues (counting multiplicities).

Problem 2   Let $\mbox{\bf {F}}$ be a subfield of a field $\mbox{\bf {K}}$. Let $p$ and $q$ be polynomials over $\mbox{\bf {F}}$. Prove that their $\gcd$ (greatest common divisor) in the ring of polynomials over $\mbox{\bf {F}}$ is the same as their $\gcd$ in the ring of polynomials over $\mbox{\bf {K}}$.

Problem 3   Let $X$ be the space of orthogonal real $n \times n$ matrices. Let $v_0 \in \mbox{$\mathbb{R}^{n}$}$. Locate and describe the elements of $X$, where the map

$$% latex2html id marker 725 f:X \to \mbox{$\mathbb{R}^{}$}, \hspace{.5in} f(A) = \langle v_0, Av_0 \rangle$$

takes its maximum and minimum values.

Problem 4   Prove that the group of automorphisms of a cyclic group of prime order $p$ is cyclic and find its order.

Problem 5  

1. Give an example of a differentiable function $f:\mbox{$\mathbb{R}^{}$}\to \mbox{$\mathbb{R}^{}$}$ whose derivative $f'$ is not continuous.

2. Let $f$ be as in Part 1. If $f'(0) < 2 < f'(1)$, prove that $f'(x) = 2$ for some $x \in [0,1]$.

Problem 6   Let $f(z) = a_0 + a_1z + \cdots + a_nz^n$ be a complex polynomial of degree $n > 0$. Prove

$$\frac{1}{2\pi i}\int_{\vert z\vert = R}z^{n-1}\vert f(z)\vert^2\,dz = a_0 \overline{a}_nR^{2n}.$$

Problem 7   Let $f$ be a continuous complex valued function on $[0,1]$, and define the function $g$ by

$$% latex2html id marker 815 g(z) = \int_0^1f(t)e^{tz}\,dt \hspace{.5in} (z\in \mbox{$\mathbb{C}\,^{}$}).$$

Prove that $g$ is analytic in the entire complex plane.

Problem 8   Let $U\subset \mbox{$\mathbb{R}^{n}$}$ be an open set. Suppose that the map
$h:U \to \mbox{$\mathbb{R}^{n}$}$ is a homeomorphism from $U$ onto $\mathbb{R}^{n}$, which is uniformly continuous. Prove $U = \mbox{$\mathbb{R}^{n}$}$.

Problem 9   Prove that a linear transformation $T : \mbox{$\mathbb{R}^{3}$} \to \mbox{$\mathbb{R}^{3}$}$ has

1. a one-dimensional invariant subspace, and

2. a two-dimensional invariant subspace.

Problem 10   Find real valued functions of a real variable, $x(t)$, $y(t)$, and $z(t)$, such that

$$x' = y, \hspace{.3in} y' = z, \hspace{.3in} z' = y$$

and

$$x(0) = 1, \hspace{.3in} y(0) = 2, \hspace{.3in} z(0) = 3.$$

Problem 11   Let $A$ and $B$ be $n\times n$ matrices over a field $\mbox{\bf {F}}$ such that $A^2=A$ and $B^2=B$. Suppose that $A$ and $B$ have the same rank. Prove that $A$ and $B$ are similar.

Problem 12   Which rational numbers $t$ are such that

$$3t^3 + 10t^2 - 3t$$

is an integer?

Problem 13   Let $\mbox{\bf {F}}$ be a finite field with $q$ elements and let $V$ be an $n$-dimensional vector space over $\mbox{\bf {F}}$.

1. Determine the number of elements in $V$.
2. Let $GL_n(\mbox{\bf {F}})$ denote the group of all $n\times n$ nonsingular matrices over $\mbox{\bf {F}}$. Determine the order of $GL_n(\mbox{\bf {F}})$.
3. Let $SL_n(\mbox{\bf {F}})$ denote the subgroup of $GL_n(\mbox{\bf {F}})$ consisting of matrices with determinant $1$. Find the order of $SL_n(\mbox{\bf {F}})$.

Problem 14   Let $A$, $B$, and $C$ be finite abelian groups such that A$\times$B and A$\times$C are isomorphic. Prove that $B$ and $C$ are isomorphic.

Problem 15   Show that

$$\sum_{n = 0}^{\infty}\frac{z}{\left(1 + z^2\right)^n}$$

converges for all complex numbers $z$ exterior to the lemniscate

$$\left\vert 1 + z^2\right\vert = 1.$$

Problem 16   Let $g_n(z)$ be an entire function having only real zeros, $n = 1, 2, \ldots$. Suppose

$$\lim_{n \to \infty}g_n(z) = g(z)$$

uniformly on compact sets in $\mathbb{C}\,^{}$, with $g$ not identically zero. Prove that $g(z)$ has only real zeros.

Problem 17   Let $f:\mbox{$\mathbb{R}^{3}$}\to\mbox{$\mathbb{R}^{}$}$ be such that

$$% latex2html id marker 1119 f^{-1}(0) = \{ x \in \mbox{$\mathbb{R}^{3}$} \;\vert\; \Vert x\Vert = 1 \}.$$

Suppose $f$ has continuous partial derivatives of orders $\leq 2$. Is there a $y \in \mbox{$\mathbb{R}^{3}$}$ with $\Vert y\Vert \leq 1$ such that

$$\frac{\partial^2f}{\partial x_1^2}(y) + \frac{\partial^2f}{\p... ... x_2^2}(y) + \frac{\partial^2f}{\partial x_3^2}(y) \geq 0 \;\;?$$

Problem 18   Let $E$ be a three-dimensional vector space over $\mathbb{Q}\,^{}$. Suppose $T:E \to E$ is a linear transformation and $Tx=y$, $Ty=z$, $Tz=x+y$, for certain $x, y, z \in E$, $x \neq 0$. Prove that $x$, $y$, and $z$ are linearly independent.

Problem 19   Let $\{f_n\}$ be a sequence of continuous functions $[0,1] \to \mbox{$\mathbb{R}^{}$}$ such that

$$\int_{0}^{1}\left(f_n(y)\right)^2\,dy \leq 5$$

for all n. Define $g_n:[0,1] \to\mbox{$\mathbb{R}^{}$}$ by

$$g_n(x) = \int_{0}^{1}\sqrt{x+y}f_n(y)\,dy.$$

1. Find a constant $K\geq 0$ such that $\vert g_n(x)\vert\leq K$ for all n.

2. Prove that a subsequence of the sequence {$g_n$} converges uniformly.

Problem 20   Let $x:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ be a solution to the differential equation

$$5x'' + 10x' + 6x = 0.$$

Prove that the function $f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$,

$$f(t) = \frac{x(t)^2}{1+x(t)^4}$$

attains a maximum value.